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Why do people care that $\pi$ and $e$ are transcendental?

user109871
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    Because that means they are not algebraic. Why would people care if a number is algebraic? Because algebraic numbers are solutions to some equation of the form $c_0+c_1x+c_2x^2+c_3x^3+\dots+c_nx^n = 0$ for some sequence of rational numbers $c_i$ and finite natural number $n$. – JMoravitz Sep 20 '18 at 16:56
  • @JMoravitz We can even assume that the $c_i$ are integers. – Peter Sep 20 '18 at 16:57
  • something to meditate on ;) – Mehness Sep 20 '18 at 16:57
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    The real question is "why do transcendental numbers matter", see my answer here. – Jack M Sep 20 '18 at 17:26

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One use for knowing whether $\pi $ is transcendental is the very old problem of squaring the circle. For thousands of years we were unable to answer it, it was only in $1882$ that we solved it, as a consequence of the Lindemann-Weierstrass theorem.

Matheus Andrade
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    +1, btw @user109871 if this motivation is interesting to you Field Theory and its Classical Problems has a truly delightful exposition of the the history of the problem leading up to the transcendence of $\pi$ in Chapter 1 The Three Greek Problems, before going onto insolvability of the quintic. (Matheus Andrade hope you don't mind tagging a reference onto your post but reminded of sth hugely enjoyed reading about decades ago!) – Mehness Sep 20 '18 at 17:17
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    @Mehness, no problem, I appreciate the suggestion as well! – Matheus Andrade Sep 20 '18 at 17:19
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    definitely worth checking out - read it for a 1st yr undergrad project on Galois theory up to Abel's Theorem, which it gets to all the way from ruler and compass in about 200 small pages :) – Mehness Sep 20 '18 at 17:30